Venn regions, set language and probability

A card is blue and striped. Let A mean blue and B mean striped. Does it belong to A only or to both sets?

Select the region in A and B

Select the region in A but not B

Select the region in neither A nor B

Read regions before totals

Circles represent sets, not separate piles. An item in the overlap belongs to both circles but is counted only once in the universe

1. 1The universe S contains every possible selected item. Each item occupies one disjoint region. The notation N(A) counts all items in A
2. 2Both and at least one An intersection means both conditions hold. A union includes either condition, including both \(\displaystyle A\cap B\quad\text{and}\quad A\cup B\)
3. 3Complement The complement of A is everything in S outside A, including B-only and neither \(\displaystyle A^{\prime}=S\setminus A\)
4. 4Only is narrower than a whole set A-only excludes B. With three sets, A-only excludes both B and C \(\displaystyle A\cap B^{\prime}\cap C^{\prime}\)
5. 5Probability needs a model If one item is selected uniformly, divide the favourable count by the universe count \(\displaystyle \Pr(E)=\frac{N(E)}{N(S)}\)

Find the total number of items in A.

Find the total number of items in S.

One of these 20 items is selected uniformly. Find the probability that it is in A or B, including both

One item is selected uniformly from S. Find the probability that it is outside A.

Which region counts are needed for neither A nor B?

Keep N(A) = 12, N(B) = 9 and N(S) = 30. Change the overlap from 0 to 9. Watch the only-regions and the outside region adjust. The circle sizes do not represent the counts

As the overlap grows while both set totals stay fixed, what happens to the union?

Universe count

\(\displaystyle 24\)

A-only / both / B-only counts

\(\displaystyle 7,\ 4,\ 5\)

Complete the missing neither-region count

For S = {1,2,3,4,5,6}, A = {1,2,4} and B = {2,3,4}, which list is A intersection B?

Sample space

\(\displaystyle S=\{1,2,3,4,5,6\}\)

Set A

\(\displaystyle A=\{1,2,4\}\)

Set B

\(\displaystyle B=\{2,3,4\}\)

One number is selected uniformly from S. Find the probability of A or B.

Add a third set

Three sets divide S into eight disjoint regions. “A and B” includes the middle of all three. “A and B only” excludes C. The word only changes the event

1. 1Two sets, excluding the third Meet A and B but not C. \(\displaystyle A\cap B\cap C^{\prime}\)
2. 2All three Meet every condition \(\displaystyle A\cap B\cap C\)

Select the region in A and C, but not B

Select the region in all three sets

Find N(A), including every region inside A.

Find the number of items in both A and B. Membership of C is allowed

One item is selected uniformly from the 20 items. Find the probability of A and B but not C.

One item is selected uniformly from the 20 items. Find the probability of exactly two of the three properties